We know that proving triangular inequalities are interesting and somewhat tough, but I use a very useful theorem that is handy in problem solving.

Let be a triangle with sides . Let be its incentre and let the incircle touch the sides on respectively. Then we have (from showing congruences): . Refer to the image.

So, we have that for some .

**PROOF**

Refer to the image.

Join .

Now, in and

is common.

Hence, from the RHS rule of congruency these two are congruent.

and are congruent , giving

Similarly we have &

QED.

**APPLICATIONS**

1. In a triangle the bisectors meet at . Prove that:

(IMO 1991/1)

2. Prove that, if are the sides of a triangle, we have:

(RMO, India)

3. For any triangle with sides , Prove that we have

(proposed by Klamkin, IMO 1983/6)

Solutions to these problems are welcome.

Thanking you.

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